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Assuming you mean x^4 = 16 (x to the power of 4 equals 16)and x^3 = 8 (x to the power of 3 equals 8).

When the power is even and the right side is positive, a solution can be negative since the 2 negatives will cancel out. For any even power, it's just an extension of that, they will cancel out in pairs. For an odd power, a negative solution would give you a negative right side so it must be positive.

Thus, in x^4=16, x can be either 2 or -2 but in x^3 = 8, x can only be 2.

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No, the atmosphere is not a solution, it is a mixture of gases. Only liquids are described as solutions.

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There's only one solution. [ x = -7 ]